Biffy Clyro is a terrible name for a band. It sounds part 1930s public school nickname, part minor character in a Tolkien-esque fantasy, with its origins obscured in the smoke of stoned goofing around. "We came up with it to annoy people," says the band's singer, Simon Neil. "We always thought if someone was gonna not listen to us because of the name, we don't want them. We've never gone out of our way to look cool."

But the name only seems strange until it becomes familiar - and it is becoming increasingly so. The three-piece from Kilmarnock in Ayrshire, Scotland, currently hold a unique position in British music as the only band to convincingly straddle both hard rock and pop. In many ways, they are a proper, old-fashioned rock band: hairy, heavily tattooed (in Neil's case), and partial to mild on-stage headbanging. They have a gang of rabid fans - Team Biffy - who follow them around. They have both coverage in the hard-rock press and Radio 1 A-listed singles; their fourth album, Puzzle, has sold 110,000. Their songs pack a punch, but are full of light, shade and sophistication; in person they are articulate, even gentle.

"We're lucky to even be able to make music," says Neil. The band is completely out of step with the Scottish music scene, which is characterised largely by folk, experimental strands of indie and Franz Ferdinand wannabes. They released their debut album, Blackened Sky, in 2002 on Beggars Banquet. Two more dense, complex yet feisty albums followed, selling in increasing but only just respectable amounts. Knowing they needed a bigger budget to do justice to their fourth album, the band moved to a major label - but found themselves "at the mercy of" the resultant negotiations between Beggars Banquet and 14th Floor, a subsidiary of Warners, which dragged on for more than a year.

In the meantime, Neil and a friend, JP Reid of fellow Ayrshire band Sucioperro, decided to make "the most fun music we could possibly make to remind ourselves why we do this". As Marmaduke Duke, they recorded an album of 18 songs in three sections: one heavy rock, one entirely acoustic, one just two bass guitars. The record sold out its 4,000 copy pressing; a second album of "disco funk" is as yet unreleased.

Neil needed the distraction. In 2004, while the band were away on tour, his mother Eleanor died, aged 54. Puzzle's addictive urgency comes out of her death. "If you're writing music or doing any kind of art, you can only do what comes out of you," says Neil. "This new music is an introduction for some people; the first songs you hear are about a very sad subject. That probably bothered me more than singing and playing the songs, because obviously the more passionate you feel about your music, the easier it is to spend eight months playing a show every night. But you have to learn to switch off - if you didn't, you'd flip out. The one song I wouldn't want to switch off for is Folding Stars, which we're never going to play again."

Released as a single in July, Folding Stars is one of Puzzle's most straightforward, sublimely melodic songs. It is also painfully open. At a time when misery and depression have become a trope, a way to make dull bands seem interesting, the song is a simple, open-hearted, open-throated howl, with Neil singing his dead mother's name repeatedly in the chorus. "I knew the record company would want it as a single," he says. "It was weird. All our parents have been very supportive of us since the start." Was it difficult hearing it everywhere? "We were out of the country, so we didn't hear it on the radio."

He pauses again. "I couldn't let my dad and my brother hear the album for a long time. My dad's not really very ... artistic. He was gonna disconnect himself to a certain extent."

Has the grief become any easier to deal with? "It gets easier to switch off. To have the chance to put your feelings out there is something people in other jobs wouldn't get. You address things that maybe initially you would find hard to. You're not going to think, 'How do I feel right now?' You're going to look back on lyrics and think, 'Well, I must have been in that situation.' It's easier to see. What a fucking cliche: 'Oh, it was therapeutic, man.' But that's what life is: a series of incredible and traumatic events. This is the best year we've had; the saddest thing that can happen to you has led to our best year. It can make you feel a bit guilty."

James Johnston, the bassist and co-singer (his twin brother Ben plays drums; both are more reserved than Neil - though not on stage) interjects briefly: "Ben and I lost a couple of friends in a car accident a couple of years ago."

"It was all so condensed," Neil says. "The car accident; six months later my grandpa; then another six months later my mum. That's why the record couldn't be about anything else."

Puzzle has taken these three young men, friends since they were "seven or eight", a long way from Kilmarnock, from jobs as road diggers and roof menders, all the way to their first major stint of American touring - from, as Ben Johnston says, "going home all grubby then playing at T in the Park.

"But it also gave us a sense of realism," he says. "You have to work for what you're gonna get. We always worked hard as a band."

Biffy Clyro have become genuine international contenders. Puzzle was mixed by Andy Wallace, a master at balancing polish and detail, as shown by his work on Jeff Buckley's 1994 album, Grace. "We always talk about Puzzle as our big record, but that implies it's all sheen," Neil says. "The fact Andy could make Grace, and Reign in Blood by Slayer, Beastie Boys' Licensed to Ill, Nirvana, At the Drive-In ... it's taking everything about a band's spirit and getting it down on to tape and making it sparkle."

Puzzle rocks, but doesn't bludgeon. "It has its moments," Ben says, "but it also has moments when it's very fragile." I tell them it's nice to hear a proper rock band who aren't afraid to use a glockenspiel, and they laugh. There are strings, too, by Graham Revell, who scored the soundtrack to the film Sin City, but they are integral to the texture, rather than just grafted on. "I think [Revell] appreciated that we weren't just giving him a song with two chords," Neil says. "We needed something that throbs at this point, or is ugly at this point, and we knew that way before we recorded the album.

"To be honest," he continues, "I think we've got any guitar-based record we wanna make. We're not a cocky band, but we're pretty confident; if we decide to make any kind of record, we'll do it."

Biffy Clyro have every right to be confident. A couple of days later, they are the first of two bands supporting Queens of the Stone Age at New York's Madison Square Gardens. Despite the jeers of a hyped up-crowd that cares for little but the headliners, they play up a storm. They play, in fact, like headliners. Eleanor Neil couldn't have a better memorial.

Biffy Clyro play the Barrowland Ballroom, Glasgow (0141-552 4601), tonight, then tour. Puzzle is out now on 14th Floor.